Question

Once I've installed AD RMS on a new server and joined to the existing cluster, I'm not interested in upgrading the existing RMS
server (http://technet.microsoft.com/en-us/library/cc771019(v=ws.10).aspx),
therefore can I just turn it off?

Or do I have to follow http://technet.microsoft.com/en-us/library/jj835767(v=ws.10).aspx instead (which looks like a
whole world of pain in comparison).

That second link is overly complex (to say the least).
If on RMS v1SP2 you are using a CNAME for your SCP it's simple.
Join the 2012 ADRMS server to the cluster and it will do the migration.
Change DNS to point the CNAME to the new server.
Take down the old one whenever you feel like it.

Another option is to do a Side Migration, so you can do a bit of testing before making the switch:

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